Equations of Plane

Show that lines $x=y,z=0$ and $x+y=0,z=0$ intersect each other. Find the vector equation of the plane determined by them.

Find the vector equation of the plane which bisects the segment joining $A\left(2,3,6\right)$ and $B\left(4,3,-2\right)$ at right angle.

Find the vector equation of the plane passing through the origin and containing the line $\overrightarrow{r}=\left(\hat{i}+4\hat{j}+\hat{k}\right)+\lambda \left(\hat{i}+2\hat{j}+\hat{k}\right)$

Find the vector equation of the plane which makes equal non-zero intercepts on the co-ordinates axes and passes through $\left(1,1,1\right)$.

Find the vector equations of planes which pass through $A\left(1,2,3\right), B\left(3,2,1\right)$ and make equal intercepts on the co-ordinates axes.

Find the Cartesian equation of the plane $\overrightarrow{r}=\lambda \left(\hat{i}+\hat{j}-\hat{k}\right)+\mu \left(\hat{i}+2\hat{j}+3\hat{k}\right)$

Find the vector equation of the plane passing through the point $A\left(-2,3,5\right)$ and parallel to vectors $4\hat{i}+3\hat{k}$ and $\hat{i}+\hat{j}$

A plane makes non zero intercepts $a,b,c$ on the co-ordinates axes. Show that the vector equation of the plane is $\overrightarrow{r}·\left(bc\hat{i}+ca\hat{j}+ab\hat{k}\right)=abc$

The foot of the perpendicular drawn from the origin to a plane is $M\left(1,2,0\right)$. Find the vector equation of the plane.

Find the Cartesian equation of the plane passing through $A\left(7,8,6\right)$ and parallel to the plane $\overrightarrow{r}·\left(6\hat{i}+8\hat{j}+7\hat{k}\right)=0$

Find the Cartesian equation of the plane passing through $A\left(1,-2,3\right)$ and the direction ratios of whose normal are $0,2,0$.

Reduce the equation $\overrightarrow{r}·\left(6\hat{i}+8\hat{j}+24\hat{k}\right)=13$ to normal form and hence find direction cosines of the normal.

Reduce the equation $\overrightarrow{r}·\left(6\hat{i}+8\hat{j}+24\hat{k}\right)=13$ to normal form and hence find the length of the perpendicular from the origin to the plane

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $2x+3y+6z=49$.

Find the vector equation of the plane which is at a distance of $5$ unit from the origin and which is normal to the vector $2\hat{i}+\hat{j}+2\hat{k}$

If the line $\frac{x+1}{2}=\frac{y-m}{3}=\frac{z-4}{6}$ lies in the plane $3x-14y+6z+49=0$, then the value of $m$ is:

The equation of the plane passing through the points $\left(1,-1,1\right),\left(3,2,4\right)$ and parallel to $Y$-axis is :

The direction cosines of the normal to the plane $2x-y+2z=3$ are

The equation of the plane passing through $\left(2,-1,3\right)$ and making equal intercepts on the coordinate axes is

Find the distance of the point $\left(1,1,-1\right)$ from the plane $3x+4y-12z+20=0$.